Abstract

We outline a full-vectorial three-dimensional multi-mode matching technique in a cylindrical coordinate system that addresses the mutual coupling among multiple modes co-propagating in a perturbed whispering gallery mode microcavity. In addition to its superior accuracy in respect to our previously implemented single-mode matching technique, this current technique is suitable for modelling waveguide-to-cavity coupling where the influence of multi-mode coupling is non-negligible. Using this methodology, a robust scheme for hybrid integration of a microcavity onto a silicon-on-insulator platform is proposed.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  25. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).
  26. S. Spillane, T. J. Kippenberg, O. J. Painter, K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
    [CrossRef] [PubMed]
  27. H. Derudder, F. Olyslager, D. De Zutter, S. Van Den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas. Propag. 49, 185–195 (2001).
    [CrossRef]

2013 (6)

2012 (2)

Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, Q. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A 85, 031805 (2012).
[CrossRef]

G. Bahl, X. Fan, T. Carmon, “Acoustic whispering-gallery modes in optomechanical shells,” New J. Phys. 14, 115026 (2012).
[CrossRef]

2011 (2)

M. Santiago-Cordoba, S. Boriskina, F. Vollmer, M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 99, 073701 (2011).
[CrossRef]

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Nat. Acad. Sci. USA 108, 5976–5979 (2011).
[CrossRef] [PubMed]

2009 (1)

Y. Sun, J. Liu, G. Frye-Mason, S.-J. Ja, A. K. Thompson, X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst 134, 1386–1391 (2009).
[CrossRef] [PubMed]

2008 (1)

F. Vollmer, S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nature Methods 5, 591–596 (2008).
[CrossRef] [PubMed]

2007 (2)

M. Oxborrow, “Traceable 2-d finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microw. Theory Tech. 55, 1209–1218 (2007).
[CrossRef]

M. Hossein-Zadeh, K. J. Vahala, “Free ultra-high-Q microtoroid: a tool for designing photonic devices,” Opt. Express 15, 166–175 (2007).
[CrossRef] [PubMed]

2006 (1)

V. Ilchenko, A. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006).
[CrossRef]

2005 (1)

2003 (4)

D. Armani, T. Kippenberg, S. Spillane, K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef] [PubMed]

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53 (2003).
[CrossRef]

I. Teraoka, S. Arnold, F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B 20, 1937–1946 (2003).
[CrossRef]

S. Spillane, T. J. Kippenberg, O. J. Painter, K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

2001 (1)

H. Derudder, F. Olyslager, D. De Zutter, S. Van Den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas. Propag. 49, 185–195 (2001).
[CrossRef]

1999 (1)

1993 (1)

D. Rowland, J. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Optoelectronics, IEEE Proceedings J 140, 177–188 (1993).
[CrossRef]

1965 (1)

Armani, D.

D. Armani, T. Kippenberg, S. Spillane, K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef] [PubMed]

Arnold, S.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Letters 13, 3347–3351 (2013).
[CrossRef] [PubMed]

F. Vollmer, S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nature Methods 5, 591–596 (2008).
[CrossRef] [PubMed]

I. Teraoka, S. Arnold, F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B 20, 1937–1946 (2003).
[CrossRef]

Bahl, G.

G. Bahl, X. Fan, T. Carmon, “Acoustic whispering-gallery modes in optomechanical shells,” New J. Phys. 14, 115026 (2012).
[CrossRef]

Barbre, C.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Letters 13, 3347–3351 (2013).
[CrossRef] [PubMed]

Bartal, G.

Bass, M.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, E. Van Stryland, Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).

Boriskina, S.

M. Santiago-Cordoba, S. Boriskina, F. Vollmer, M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 99, 073701 (2011).
[CrossRef]

Borselli, M.

Carmon, T.

Chen, T.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Nat. Acad. Sci. USA 108, 5976–5979 (2011).
[CrossRef] [PubMed]

Chen, Y.-L.

Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, Q. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A 85, 031805 (2012).
[CrossRef]

Cohen, O.

Dantham, V. R.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Letters 13, 3347–3351 (2013).
[CrossRef] [PubMed]

De Zutter, D.

H. Derudder, F. Olyslager, D. De Zutter, S. Van Den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas. Propag. 49, 185–195 (2001).
[CrossRef]

DeCusatis, C.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, E. Van Stryland, Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).

Demirel, M.

M. Santiago-Cordoba, S. Boriskina, F. Vollmer, M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 99, 073701 (2011).
[CrossRef]

Derudder, H.

H. Derudder, F. Olyslager, D. De Zutter, S. Van Den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas. Propag. 49, 185–195 (2001).
[CrossRef]

Du, X.

Enoch, J.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, E. Van Stryland, Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).

Fan, X.

G. Bahl, X. Fan, T. Carmon, “Acoustic whispering-gallery modes in optomechanical shells,” New J. Phys. 14, 115026 (2012).
[CrossRef]

Y. Sun, J. Liu, G. Frye-Mason, S.-J. Ja, A. K. Thompson, X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst 134, 1386–1391 (2009).
[CrossRef] [PubMed]

Flagan, R. C.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Nat. Acad. Sci. USA 108, 5976–5979 (2011).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).

Foreman, M. R.

M. R. Foreman, F. Vollmer, “Theory of resonance shifts of whispering gallery modes by arbitrary plasmonic nanoparticles,” New J. Phys. 15, 083006 (2013).
[CrossRef]

Fraser, S. E.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Nat. Acad. Sci. USA 108, 5976–5979 (2011).
[CrossRef] [PubMed]

Frye-Mason, G.

Y. Sun, J. Liu, G. Frye-Mason, S.-J. Ja, A. K. Thompson, X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst 134, 1386–1391 (2009).
[CrossRef] [PubMed]

Gong, Q.

Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, Q. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A 85, 031805 (2012).
[CrossRef]

Gorodetsky, M. L.

Guo, G.-C.

Haus, H. A.

H. A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000).
[CrossRef]

Herchak, S.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Nat. Acad. Sci. USA 108, 5976–5979 (2011).
[CrossRef] [PubMed]

Holler, S.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Letters 13, 3347–3351 (2013).
[CrossRef] [PubMed]

Hossein-Zadeh, M.

Ilchenko, V.

V. Ilchenko, A. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006).
[CrossRef]

Ilchenko, V. S.

Ja, S.-J.

Y. Sun, J. Liu, G. Frye-Mason, S.-J. Ja, A. K. Thompson, X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst 134, 1386–1391 (2009).
[CrossRef] [PubMed]

Johnson, T.

Kaplan, A.

Keng, D.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Letters 13, 3347–3351 (2013).
[CrossRef] [PubMed]

Kim, J.-H.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Nat. Acad. Sci. USA 108, 5976–5979 (2011).
[CrossRef] [PubMed]

Kippenberg, T.

D. Armani, T. Kippenberg, S. Spillane, K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef] [PubMed]

Kippenberg, T. J.

S. Spillane, T. J. Kippenberg, O. J. Painter, K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Kolchenko, V.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Letters 13, 3347–3351 (2013).
[CrossRef] [PubMed]

Kozlov, M.

Lakshminarayanan, V.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, E. Van Stryland, Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).

Lee, H.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Nat. Acad. Sci. USA 108, 5976–5979 (2011).
[CrossRef] [PubMed]

Li, B.-B.

Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, Q. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A 85, 031805 (2012).
[CrossRef]

Li, G.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, E. Van Stryland, Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).

Li, Y.

Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, Q. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A 85, 031805 (2012).
[CrossRef]

Liu, J.

Y. Sun, J. Liu, G. Frye-Mason, S.-J. Ja, A. K. Thompson, X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst 134, 1386–1391 (2009).
[CrossRef] [PubMed]

Liu, Y.-C.

Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, Q. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A 85, 031805 (2012).
[CrossRef]

Love, J.

D. Rowland, J. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Optoelectronics, IEEE Proceedings J 140, 177–188 (1993).
[CrossRef]

Lu, T.

Macdonald, C.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, E. Van Stryland, Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).

Mahajan, V.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, E. Van Stryland, Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).

Malitson, I. H.

Matsko, A.

V. Ilchenko, A. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006).
[CrossRef]

Olyslager, F.

H. Derudder, F. Olyslager, D. De Zutter, S. Van Den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas. Propag. 49, 185–195 (2001).
[CrossRef]

Oxborrow, M.

M. Oxborrow, “Traceable 2-d finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microw. Theory Tech. 55, 1209–1218 (2007).
[CrossRef]

Painter, O.

Painter, O. J.

S. Spillane, T. J. Kippenberg, O. J. Painter, K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).

Ren, X.-F.

Rowland, D.

D. Rowland, J. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Optoelectronics, IEEE Proceedings J 140, 177–188 (1993).
[CrossRef]

Santiago-Cordoba, M.

M. Santiago-Cordoba, S. Boriskina, F. Vollmer, M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 99, 073701 (2011).
[CrossRef]

Schwefel, H. G. L.

Shirazi, M. A. C.

Spillane, S.

S. Spillane, T. J. Kippenberg, O. J. Painter, K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

D. Armani, T. Kippenberg, S. Spillane, K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef] [PubMed]

Sun, Y.

Y. Sun, J. Liu, G. Frye-Mason, S.-J. Ja, A. K. Thompson, X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst 134, 1386–1391 (2009).
[CrossRef] [PubMed]

Teraoka, I.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).

Thompson, A. K.

Y. Sun, J. Liu, G. Frye-Mason, S.-J. Ja, A. K. Thompson, X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst 134, 1386–1391 (2009).
[CrossRef] [PubMed]

Tomes, M.

Vahala, K.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Nat. Acad. Sci. USA 108, 5976–5979 (2011).
[CrossRef] [PubMed]

D. Armani, T. Kippenberg, S. Spillane, K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef] [PubMed]

Vahala, K. J.

M. Hossein-Zadeh, K. J. Vahala, “Free ultra-high-Q microtoroid: a tool for designing photonic devices,” Opt. Express 15, 166–175 (2007).
[CrossRef] [PubMed]

S. Spillane, T. J. Kippenberg, O. J. Painter, K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Van Den Berghe, S.

H. Derudder, F. Olyslager, D. De Zutter, S. Van Den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas. Propag. 49, 185–195 (2001).
[CrossRef]

Van Stryland, E.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, E. Van Stryland, Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).

Vincent, S.

Vollmer, F.

M. R. Foreman, F. Vollmer, “Theory of resonance shifts of whispering gallery modes by arbitrary plasmonic nanoparticles,” New J. Phys. 15, 083006 (2013).
[CrossRef]

M. Santiago-Cordoba, S. Boriskina, F. Vollmer, M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 99, 073701 (2011).
[CrossRef]

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Supplementary Material (1)

» Media 1: MP4 (11296 KB)     

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Figures (2)

Fig. 1:
Fig. 1:

(a) Geometry of the toroid-SOI system and top view of the electrical field distribution. The accompanying propagation animation, Media 1, exhibits SOI waveguide width variation in the inset which is attributed to the progression of the azimuthal cross sections along the cavity’s circumference. (b) The amplitude of the two lowest-order WGM’s along the azimuthal direction when a straight SOI waveguide is placed at the equator of the cavity (i.e. leftmost inset). The transverse electric field distributions of the launched mode at ϕ = −0.034 rad (left) and ϕ = 0 rad (right) are displayed as insets. (c) The reciprocal of the quality factor as a function of azimuthal angle step size, indicating a convergence error of Oϕ1.0) from the least square fitting of the last three, smallest Δϕ. (d) The number of modes included in the simulation, revealing a convergence rate of around O(N0.8) via the least square fitting of the relative error (excluding the N = 1 point).

Fig. 2:
Fig. 2:

Coupling Q-factor as a function of (a) gap size and (b) vertical angle. (c) Coupling parameter K (i.e. ratio between the power of the waveguide mode and the power lost within the entire system) versus the gap size for a 67 μm-diameter microsphere and 1.35 μm-diameter fiber at λ =1550 nm.

Equations (10)

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0 2 π d ϕ d z 0 ε ( ρ , z , ϕ ) 2 [ e j m ˜ l ϕ e ˜ ^ M , l * ( ρ , z ) ] [ e j m ˜ k ϕ e ˜ ^ M , k ( ρ , z ) ] ρ d ρ = δ l k
e ˜ ^ l ( ρ , z ) | e ˜ ^ k ( ρ , z ) = π d z 0 ε ( ρ , z ) e ˜ ^ l * ( ρ , z ) e ˜ ^ k ( ρ , z ) ρ d ρ = δ l k
E ( ρ , z , ϕ 0 ) = l = 1 N ( ϕ 0 ) A l ( ϕ 0 ) | e ˜ ^ l ( ρ , z , ϕ 0 )
E ( ρ , z , ϕ 0 + δ ϕ 0 ) = l = 1 N ( ϕ 0 ) A l ( ϕ 0 ) e j m l ( ϕ 0 , λ ) δ ϕ 0 | e ˜ ^ l ( ρ , z , ϕ 0 )
E ( ρ , z , ϕ 0 + δ ϕ 0 ) = l = 1 N ( ϕ 0 + δ ϕ 0 ) A l ( ϕ 0 + δ ϕ 0 ) | e ˜ ^ l ( ρ , z , ϕ 0 + δ ϕ 0 )
A ( ϕ 0 + δ ϕ 0 ) = C ˜ ( ϕ 0 ) A ( ϕ 0 )
{ C ˜ ( ϕ 0 ) } k l = e [ j m l ( ϕ 0 , λ ) m k l ] δ ϕ 0
A ( ϕ 0 + 2 π ) = p = 0 P 1 C ˜ ( ϕ p , λ ) A ( ϕ 0 )
| e ^ p ( ρ , z , ϕ 0 ) = q A p , q ( ϕ 0 ) | e ˜ ^ q ( ρ , z , ϕ 0 )
E ( ρ , z , ϕ 0 + δ ϕ ) = k = 1 N ( ϕ 0 ) A l ( ϕ 0 ) e j ϕ 0 ϕ 0 + δ ϕ m l ( ϕ , λ ) d ϕ | e ˜ ^ l ( ρ , z , ϕ 0 )

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